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Mirrors > Home > MPE Home > Th. List > eldprdi | Structured version Visualization version GIF version |
Description: The domain of definition of the internal direct product, which states that 𝑆 is a family of subgroups that mutually commute and have trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.) |
Ref | Expression |
---|---|
eldprdi.0 | ⊢ 0 = (0g‘𝐺) |
eldprdi.w | ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } |
eldprdi.1 | ⊢ (𝜑 → 𝐺dom DProd 𝑆) |
eldprdi.2 | ⊢ (𝜑 → dom 𝑆 = 𝐼) |
eldprdi.3 | ⊢ (𝜑 → 𝐹 ∈ 𝑊) |
Ref | Expression |
---|---|
eldprdi | ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldprdi.1 | . 2 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
2 | eldprdi.3 | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑊) | |
3 | eqid 2738 | . . 3 ⊢ (𝐺 Σg 𝐹) = (𝐺 Σg 𝐹) | |
4 | oveq2 7283 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝐺 Σg 𝑓) = (𝐺 Σg 𝐹)) | |
5 | 4 | rspceeqv 3575 | . . 3 ⊢ ((𝐹 ∈ 𝑊 ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝐹)) → ∃𝑓 ∈ 𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)) |
6 | 2, 3, 5 | sylancl 586 | . 2 ⊢ (𝜑 → ∃𝑓 ∈ 𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)) |
7 | eldprdi.2 | . . 3 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
8 | eldprdi.0 | . . . 4 ⊢ 0 = (0g‘𝐺) | |
9 | eldprdi.w | . . . 4 ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } | |
10 | 8, 9 | eldprd 19607 | . . 3 ⊢ (dom 𝑆 = 𝐼 → ((𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)))) |
11 | 7, 10 | syl 17 | . 2 ⊢ (𝜑 → ((𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)))) |
12 | 1, 6, 11 | mpbir2and 710 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∃wrex 3065 {crab 3068 class class class wbr 5074 dom cdm 5589 ‘cfv 6433 (class class class)co 7275 Xcixp 8685 finSupp cfsupp 9128 0gc0g 17150 Σg cgsu 17151 DProd cdprd 19596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-ixp 8686 df-dprd 19598 |
This theorem is referenced by: dprdfsub 19624 dprdf11 19626 dprdsubg 19627 dprdub 19628 dpjidcl 19661 |
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